# Divergence of first Piola-Kirchoff stress tensor

• I
• bobinthebox
In summary, the conversation discusses the first Piola-Kirchoff stress tensor and the definition of divergence for a tensor field in the case of studying the bending of an incompressible elastic block. The equation for the first Piola-Kirchoff stress tensor is given as equation 5.93 on page 182, with the terms $e_r$ and $e_{\theta}$ defined in terms of $\theta$. The definition of divergence for $S$ is given as $Div(S) = \frac{\partial }{\partial x_1^0} S_{r1} e_r + \frac{\partial}{\partial x_2^0} S_{\theta 2} e_{\theta} bobinthebox TL;DR Summary How is the divergence defined for such a tensor field? Hi everyone, studying the bending of an incompressible elastic block of Neo-Hookean material, one finds out the first Piola-Kirchoff stress tensor as at page 182 (equation 5.93) where$e_r = cos(\theta)e_1 + \sin(\theta)e_2$and$e_{\theta} = -sin(\theta)e_1 + \cos(\theta)e_2$How is the divergence defined in this case? The book writes the following $$Div(S) = \frac{\partial }{\partial x_1^0} S_{r1} e_r + \frac{\partial}{\partial x_2^0} S_{\theta 2} e_{\theta} = 0$$ but I can't see why this is the divergence of$S$. I'd like to have a formal proof of this. I know that, by definition, the divergence of a tensor field$S$is the unique vector field such that$Div(S^T a)=Div(S) \cdot a$for every vector$a$and$S^T$is the transpose of$S\$. But I don't know how to apply this in this particular case.Thanks a lot!

## 1. What is the first Piola-Kirchoff stress tensor?

The first Piola-Kirchoff stress tensor is a mathematical quantity used in continuum mechanics to describe the stress state of a deformable material. It is a second-order tensor that relates the forces acting on a material to its deformation.

## 2. How is the first Piola-Kirchoff stress tensor different from the Cauchy stress tensor?

The Cauchy stress tensor describes the stress state of a material in its current configuration, while the first Piola-Kirchoff stress tensor describes the stress state in the material's reference configuration. This means that the first Piola-Kirchoff stress tensor takes into account the initial orientation and shape of the material, while the Cauchy stress tensor only considers the current state.

## 3. What is the significance of the divergence of the first Piola-Kirchoff stress tensor?

The divergence of the first Piola-Kirchoff stress tensor is a measure of the rate of change of stress in a material. It is important in understanding the behavior of materials under different loading conditions and can be used to predict the deformation and failure of structures.

## 4. How is the divergence of the first Piola-Kirchoff stress tensor calculated?

The divergence of the first Piola-Kirchoff stress tensor is calculated by taking the derivative of each component of the tensor with respect to the corresponding coordinate direction. This results in a second-order tensor known as the material stress-strain tensor.

## 5. What are some applications of the first Piola-Kirchoff stress tensor?

The first Piola-Kirchoff stress tensor is used in various fields such as solid mechanics, fluid mechanics, and biomechanics. It is important in the analysis and design of structures and materials, as well as in understanding the behavior of biological tissues. It is also used in numerical simulations and finite element analysis to predict the response of materials under different loading conditions.

• Mechanics
Replies
4
Views
839
• Mechanical Engineering
Replies
21
Views
1K
• Mechanics
Replies
10
Views
912
• Mechanics
Replies
8
Views
877
• Differential Geometry
Replies
3
Views
2K
• Classical Physics
Replies
17
Views
10K
Replies
3
Views
3K
• Calculus
Replies
2
Views
1K